First slide

Methods of integration

Question

Evaluate $\int x^{\frac{- 1}{3}} {(1 + x^{1 / 3})}^{1 / 4} dx$

Moderate

A

$\frac{- 4}{15} {(1 + x^{1 / 3})}^{5 / 4} (4 + 9 x^{1 / 2}) + C$
B

$\frac{1}{15} {(1 + x^{1 / 3})}^{5 / 4} (4 + 9 x^{1 / 3}) + C$
C

$\frac{2}{15} {(1 + x^{1 / 3})}^{1 / 4} (4 + 9 x^{1 / 3}) + C$
D

None of the above

Solution

Here, $m = \frac{- 1}{3}, n = \frac{1}{3}, p = \frac{1}{4}, \frac{m + 1}{n} = 2$
Which is an integer.
So, $(1 + x^{1 / 3}) = t^{4}$
$\begin{array}{l} \Rightarrow \frac{dx}{3 x^{2 / 3}} = 4 t^{3} dt \\ I = 12 \int (t^{4} - 1) t^{4} dt \\ I = 12 [t^{8} - t^{4} \cdot dt] \\ = 12 [\frac{t^{9}}{9} - \frac{t^{5}}{5}] + C [∵ C = 12 C_{1}] \\ = 4 t^{5} [\frac{t^{4}}{3} - \frac{3}{5}] + C \\ = \frac{4}{15} {(1 + x^{1 / 3})}^{5 / 4} \cdot [5 x^{1 / 3} - 4] \end{array}$

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